3. Prove that coshx – coshy = 2sinh () sinh ().

Q: dy +y = zy dr

A: Given equation is: xdydx+y=xy3

Q: Solve the following DEQ by separable method: dy (cos x + 1)( sin y – 1) dx

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Q: 2. Evaluate the integral / sin'at dt

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Q: 5. Evalvate the Intgral S2 cos (2x) - Sin 2x)=

A: To evaluate the integral: ∫2cos22x·sin22x dx Solution: We know that, 1). 2sinθ cosθ=sin2θ 2).…

Q: 4) a²y=+52y =14 yat (2,1) Find equátion of tangent?

A: Equation of tangent  y-y1=m(x-x1) (x1,y1)are the points at which equation of tangent is to be…

Q: Prove that sinh−1 t = ln(t + √t2 + 1 ).

A:

Q: 15. Use a suitable change of variables to determine the indefinite integral. [1²cos² (1³) sin(1³)dt

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Q: 10. Use differentials to estimate to estimate sin(46°). vZ(n+180) А. D. VZ(T+180) 360 180 В.…

A: Let y = sin (x)

Q: 4ス+S fra) = 7 +2 Contromulily of he funihim ihe fumnim Discuss The at x= 1

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Q: If xy In(3 sinh( 3x)), then x2y'= %3D Select one: Ox coth( 3x ) - In( 3 sinh(3x)) 09x coth( 3x ) - 3…

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Q: to tal differential of eind the the り23 function ] 2=e + tan Z= Sin 2.

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Q: Q3. Show that дхду for дудх f = x² siny – 2y³ cos2x (

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Q: 4. Show that 2 s w (b) L(t sinh (wt)) (s² – w²)2 -

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Q: Q2) prove that cosh(x + y) = coshx coshy + sinhx sinhy

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Q: Estimate the error if cos t² is approximated by 1 integral , cos r² dt. 4! in the 2

A: We have,

Q: )L{sinh(7t) + (2t + 1)U (t – 1)},

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Q: Given: 4 2 y = -arctan sinh 3 х — find y' at x = 2 - cosh 1 - y 1 + sinh? 1 %3D cosh 1 y = 1 + sinh…

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Q: Evaluate the integral: S/2 sin(4t - u) du dt

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Q: If Z= sinh () show that: *=+y==0

A: GivenZ=sinhyx

Q: Prove that sinh−1(x) = In( x+Vx2 +1)

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Q: Evaluate the integral. sin (3t + 17) dt cos (3t + 17) sin (3t + 17) dt = - 2, cos (3t + 17)

A: Consider the integral

Q: If r = (3 cosh 2t)i+ (3 sinh 2t)j+ 6tk and t = In 2 then اخترأحد الخيارات

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Q: dy dr

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Q: From the differential equati on by eliminating the arbitrary function ffrom xyz = f (x?+y'+z°).

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Q: show that the differential eguation is exact, then solre that method using 2 2 dy- x²-x+y? |

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Q: is 4= Sinh(2sinhix) shaw Hh at %3D 2 (a)

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Q: Consider f(x) = sin(2x) + cos(4x), 0 ≤ x ≤ 1. .......Integrate f over the given interval of x, using…

A: Solution:

Q: hD (Hint: E =eD and Sinh x: 2 Q3: Prove that 8 = 2Sinh %3D

A: We will prove the following result.

Q: 4. Get the exact a. dy = cos 2x dz differentials of y = (sin 2)² b. dy=sin 2z dz dy = sin 2x d.…

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Q: Evaluate the integral 3 2x cos(xy)dydx 2

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Q: -Tsin(nx)cosh(3y) dx + 3cos(7x)sinh(3y) dy =0

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Q: 10 a) Evaluate x cos(5x) dx by using integration by parts.

A: Given:

Q: This is exercise 12 of Section 7.5 page 460. Find the integral sin(2x) dx +C 1+ sin (x)

A:

Q: 2. Use the substitution technique to find [*5'sin(5¹) dt exactly (not an approximation).

A: The given integral is ∫015tsin5tdt use the formulas, ddtat=atln(a)∫sint dt=-cost+C

Q: Hico/ prove Thall Cosh2: - 2 tañ z Sinhz: 1+2 Tanhiz =

A: Since you have asked for a question with multiple sub-parts , we will solve the first three…

Q: I Braluate th FOllowing integralis 3+cos (logqx) dx x sin (loga x )

A:

Q: Use the methods of differentials to estimate the value of (x+1) over the square root of 2x+1 at…

A: Use the quotient rule to find the derivative for f(x).

Q: 2π 3 [³* sin(y) + sec² (y) dy

A: We have to evaluate the given integral.

Q: if t(い) = ta Cosy V = X siny Determine JR ļ JR to つメ

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Q: Evaluate the integral: sinh 3x csch 3x dx coth 3x "+C 3 coth X +C 3 - coth x +C - 3 - coth 3x +C 3

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Q: 14. Use a suitable change of variables to determine the indefinite integral. fsin²xcos³ xdx (Hint:…

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Q: ons z = sin y arn

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Q: B ) if y= sinh¬1x then prove that =

A: Given y = sinh-1(x)sinh(y) = x On differentiating both sides , we get sinh(y)' = (x)'(cosh(y))dydx…

Q: y²

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Q: (4z + 5)² tan zy= + dr Cos z

A:

Q: b) Use it in oonjunction with Simpson's 1/3 nule to integrate the same function for ive egmenta…

A: Introduction: Simpson's rule is a method to approximate a definite integral. The interval must be…

Q: b) Find if e Vt2 +5 dt + siny = x- xcosy rsinx ycscx

A: According to Leibniz' rule:

Q: B) Prove that: cosh (x + y) = coshx coshy + sinhx sinhy %3D

A: We know that cosh(θ) = eθ+e-θ2sinh(θ) = eθ-e-θ2

Q: Find Z in (1-j5) 2= tanh cos lat5 express in lagarithm Form

A:

Minimization

In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.

Maxima and Minima

The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.

Derivatives

A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.  

Concavity

In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.